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Here is what the theory said :

Here is the Barkhausen stability criterion :

The loop gain is equal to unity in absolute magnitude, that is,

$$|\beta A|= 1 $$

and the phase shift around the loop is zero or an integer multiple of 2π:

$$\angle{\beta A} = 2\pi n, n \in 0,1,2... $$

Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate

Which means also that when the criterion is met :

$$\frac{A}{1+AB} = \infty$$

Well, I did some simulations. Actually, I was thinking that if I have a positive feedback with a gain superior to 1, it will oscillate. Mathematically It means :

$$|\beta A| \geq 1 $$

$$\angle{\beta A} = 2\pi n, n \in 0,1,2... $$

But it ends with the closed loop transfer function which is no longer equal to infinity.

$$\frac{A}{1+AB} \neq \infty$$

Here are the simulation that I have done and the simulation makes me think that I am right, so what I did not understand ?

Firstly here is a system which respect the Barkhausen criteria :

Bode diagram open loop : enter image description here

Bode diagram Closed loop: enter image description here

Step response to 1V : enter image description here

The circuit made for the simulation : enter image description here

Here is the second circuit with a gain higher than 1 @ -180° : Bode diagram open loop : enter image description here

Bode diagram Closed loop with the previous system in green : enter image description here

Step response to 1V : enter image description here

The circuit made for the simulation : enter image description here

As you can see the 2nd case is completely more unstable than the first one and the closed loop transfer function does not go to infinity as the Barkhausen Criterion is not respected.

Thank you :)

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There is a fundamental error in your analysis:

When you write the closed loop gain Acl as

Acl=A/(1+AB) with feedback factor B and open-loop gain A

you are assuming that there is a NEGATIVE sign at the summing junction (where the feedback signal is added). This is identical to negative fedback.

In this case, the oscillation criterion from Barkhausen (it should not be called " Stability Criterion") requires an additional phase shift (caused by the external RC network) at the desired oscillation frequency wo of 180deg only because of the additional phase inversion at the summing junction.

The original criterion is based on positive feedback (at w=wo).That means: If you want to write down the closed loop gain the following expression applies:

Acl=A/(1-AB).

And - as you can see - the denominator is approaching zero for AB=1 (unity magnitude and zero resp. 360 deg phase shift). Note that the product AB is identical to the gain of the whole feedback loop (including all phase shifts) - and it is called "loop gain".

Therefore, in short: Barkhausen criterion requires unity loop gain at w=wo. (And you are right: It is a necessary oscillation criterion only).

However, a working oscillator needs negative feedback at DC (w=0).

But - for which purpose do you want to investigate the closed-loop gain expression? An oscillator has no external input signal to be amplified.

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